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- Jouni Hartikainen, Simo Sarkka
- 2010 IEEE International Workshop on Machine…
- 2010

In this paper, we show how temporal (i.e., time-series) Gaussian process regression models in machine learning can be reformulated as linear-Gaussian state space models, which can be solved exactly with classical Kalman filtering theory. The result is an efficient non-parametric learning algorithm, whose computational complexity grows linearly with respect… (More)

- Jarno Vanhatalo, Jaakko Riihimäki, Jouni Hartikainen, Pasi Jylänki, Ville Tolvanen, Aki Vehtari
- Journal of Machine Learning Research
- 2013

The GPstuff toolbox is a versatile collection of Gaussian process models and computational tools required for Bayesian inference. The tools include, among others, various inference methods, sparse approximations and model assessment methods.

- Robert Piché, Simo Särkkä, Jouni Hartikainen
- 2012 IEEE International Workshop on Machine…
- 2012

Nonlinear Kalman filter and Rauch-Tung-Striebel smoother type recursive estimators for nonlinear discrete-time state space models with multivariate Student's t-distributed measurement noise are presented. The methods approximate the posterior state at each time step using the variational Bayes method. The nonlinearities in the dynamic and measurement models… (More)

- Simo Särkkä, Arno Solin, Jouni Hartikainen
- IEEE Signal Processing Magazine
- 2013

Gaussian process-based machine learning is a powerful Bayesian paradigm for nonparametric nonlinear regression and classification. In this article, we discuss connections of Gaussian process regression with Kalman filtering and present methods for converting spatiotemporal Gaussian process regression problems into infinite-dimensional state-space models.… (More)

- Simo Särkkä, Jouni Hartikainen
- AISTATS
- 2012

We show how spatio-temporal Gaussian process (GP) regression problems (or the equivalent Kriging problems) can be formulated as infinite-dimensional Kalman filtering and Rauch-Tung-Striebel (RTS) smoothing problems, and present a procedure for converting spatio-temporal covariance functions into infinite-dimensional stochastic differential equations (SDEs).… (More)

- Jouni Hartikainen, Simo Särkkä
- UAI
- 2011

Latent force models (LFMs) are hybrid models combining mechanistic principles with non-parametric components. In this article, we shall show how LFMs can be equivalently formulated and solved using the state variable approach. We shall also show how the Gaussian process prior used in LFMs can be equivalently formulated as a linear statespace model driven by… (More)

- Simo Särkkä, Jouni Hartikainen
- IEEE Trans. Automat. Contr.
- 2010

In this note we shall present a new Gaussian approximation based framework for approximate optimal smoothing of non-linear stochastic state space models. The approximation framework can be used for efficiently solving non-linear fixedinterval, fixed-point and fixed-lag optimal smoothing problems. We shall also numerically compare accuracies of… (More)

Gaussian processes (GP) are powerful tools for probabilistic modeling purposes. They can be used to define prior distributions over latent functions in hierarchical Bayesian models. The prior over functions is defined implicitly by the mean and covariance function, which determine the smoothness and variability of the function. The inference can then be… (More)

- Simo Särkkä, Jouni Hartikainen
- IEEE Trans. Automat. Contr.
- 2011

- Jouni Hartikainen, Mari Seppänen, Simo Särkkä
- ICML
- 2012

Latent force models (LFMs) are flexible models that combine mechanistic modelling principles (i.e., physical models) with nonparametric data-driven components. Several key applications of LFMs need nonlinearities, which results in analytically intractable inference. In this work we show how non-linear LFMs can be represented as nonlinear white noise driven… (More)